President Donald Trump opened a new flank in his battle against illegal immigration on Monday when he ordered his administration to crack down on "visa overstays" – foreigners who legally enter the country but remain in the U.S. after their visas expire.

The president signed a memorandum ordering the secretary of state and the secretary of homeland security to submit plans within four months to crack down on overstays, such as punishing countries whose citizens have high rates of overstays and requiring foreign travelers to post "admission bonds" that would be repaid once they leave the country.

Members of both parties have long complained that overstays are just as problematic as undocumented immigrants who cross the southern border. More than 1.2 million foreigners overstayed their visas from 2016 to 2017, according to the most recent Homeland Security data.

"Although the United States benefits from legitimate (non-immigrant) entry, individuals who abuse the visa process and decline to abide by the terms and conditions of their visas, including their visa departure dates, undermine the integrity of our immigration system and harm the national interest," Trump wrote in his memorandum.

The federal government only recently started closely tracking visa overstays, issuing its first report in January 2016 during the Obama administration. But the reports immediately revealed the scope of the problem – more than 600,000 foreigners overstayed their visas each year from 2016 to 2017.

And with illegal immigration remaining at historic lows, that means the majority of new undocumented immigrants are legally entering through land, air and sea ports with visas in hand.

Stopping visa overstays has proved just as difficult as sealing the southern border.

In the aftermath of the 9/11 terrorist attacks, Congress passed a law requiring that the federal government capture the fingerprints of all incoming, and departing, foreigners to better track who enters the country and exits. Such "biometric" information is considered more reliable than the "biographic" information that's currently the norm – names, birthdates and paper travel documents.

U.S. Customs and Border Protection has initiated many pilot projects to test that technology, including facial recognition software that is currently being used at 13 airports around the country. But the government is far from being able to implement the kind of nationwide, biometric exit-entry system ordered by Congress nearly 20 years ago.

Trump's memorandum offers up other suggestions for ways to crack down on overstays.

He ordered Secretary of State Mike Pompeo to "engage with the governments" of countries where more than 10% of their citizens traveling to the U.S. on short-term visas do not leave the country when their visas expire. Possible punishments include limiting the number of visas granted to citizens of those countries, limiting the time its citizens are allowed to travel to the U.S. and requiring its citizens to provide more documents when traveling.

"The Secretary of State and the Secretary of Homeland Security shall immediately begin taking all appropriate actions that are within the scope of their respective authorities to reduce overstay rates for all classes of (non-immigrant) visas," Trump wrote.